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The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the αk-generalized derivation algebra G D e r (J) of a ternary Hom-Jordan algebras. In particular, we prove that G D e r (J) = Q D e r (J) + Q Γ (J) , the sum of the αk-quasiderivation algebra and the αk-quasicentroid. We also prove that Q D e r (J) can be embedded as derivations in a larger ternary Hom-Jordan algebras. General results on αk-centroids of ternary Hom-Jordan algebras are also developed in the paper. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

Let H be a real or complex Hilbert space with the dimension greater than one and B (H) the algebra of all bounded linear operators on H . Assume that δ is a linear mapping from B (H) into itself which is Jordan derivable at a given element Ω ∈ B (H) , in the sense that δ (A ∘ B) = δ (A) ∘ B + A ∘ δ (B) holds for all A , B ∈ B (H) with A ∘ B = Ω , where ∘ denotes the Jordan product A ∘ B = A B + B A . In this paper, we show that if Ω is an arbitrary but fixed nonzero operator, then δ is a derivation; if Ω is a zero operator, then δ is a generalized derivation. [ABSTRACT FROM AUTHOR]

In this paper we consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in [AAH1], and its restricted version in odd characteristic introduced in [AAH2]. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in loc. cit. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of osp(1|2). [ABSTRACT FROM AUTHOR]

In this paper, we proved that each nonlinear non-global semi-Jordan triple higher derivable mapping on a 2-torsion free triangular algebra is an additive higher derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively. [ABSTRACT FROM AUTHOR]

It is well known that any Lie algebra can be embedded into an associative algebra. We prove that any metabelian Lie algebra can be embedded into an algebra in the subvariety of perm algebras, i.e. associative algebras with the identity a b c − a c b = 0. In addition, a technical method to construct the universal enveloping perm algebra for a metabelian Lie algebra is given. [ABSTRACT FROM AUTHOR]

We consider a class of generalized derivations that arise in connection with the problem of adjoining unity to an algebra with generalized derivation, and of searching envelopes for Novikov–Poisson algebras. Conditions for the existence of the localization of an algebra with ternary derivation are specified, as well as conditions under which given an algebra with ternary derivation, we can construct a Novikov–Poisson algebra and a Jordan superalgebra. Finally, we show how the simplicity of an algebra with Brešar generalized derivation is connected with simplicity of the appropriate Novikov algebra. [ABSTRACT FROM AUTHOR]

Let H be an infinite dimensional Hilbert space and B (H) be the algebra of all bounded linear operators on H . For A ∈ B (H) , let σ ∗ (A) denote the semi-Fredholm domain, the Fredholm domain or the Weyl domain of A in the spectrum, σ(A). We characterize the form of a map ϕ from B (H) into itself with range contains the operators of rank at most two and satisfies σ ∗ (ϕ (A) ϕ (B) ∗) = σ ∗ (A B ∗) for all A , B ∈ B (H). We also describe those maps on B (H) preserving the same spectral set σ ∗ (.) 'but of Jordan skew-triple product' A B ∗ A 'of operators'. [ABSTRACT FROM AUTHOR]

This article aims to interpret some Lie triple derivations of Tensor algebras T ⊗ F (a , b F) , using generalized quaternion algebra (a , b F) over a field F and assuming T as an F-linear associative algebra. The study concludes with complete description of all Lie triple derivations of A ⊗ FL. It is pertinent to mention here that A is assumed to be a finite-dimensional commutative algebra and L is a simple Lie algebra over an algebraically closed field. [ABSTRACT FROM AUTHOR]

We define the notion of an almost polynomial identity of an associative algebra R, and show that its existence implies the existence of an actual polynomial identity of R. A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras. [ABSTRACT FROM AUTHOR]

Let A be a unital algebra and M be a unital A -bimodule. We characterize the linear mappings δ and τ from A into M , satisfying δ (A) B + A τ (B) = 0 for every A , B ∈ A with A B = 0 when A contains a separating ideal T of M , which is in the algebra generated by all idempotents in A . We apply the result to P -subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that A is a unital Banach algebra and M is a unital Banach A -bimodule, we give a complete description of linear mappings δ and τ from A into M , satisfying δ (A) B + A τ (B) = 0 for every A , B ∈ A with A B = I . [ABSTRACT FROM AUTHOR]

We demonstrate the way to derive the second Painlevé equation P 2 and its Bäcklund transformations from the deformations of the Nonlinear Schrödinger equation (NLS), all the while preserving the strict invariance with respect to the Schlesinger transformations. The proposed algorithm allows for a construction of Jordan algebra-based completely integrable multiple-field generalizations of P 2 while also producing the corresponding Bäcklund transformations. We suggest calling such models the JP-systems. For example, a Jordan algebra J Mat (N , N) with the Jordan product in the form of a semi-anticommutator is shown to generate an integrable matrix generalization of P 2 , whereas the V N algebra produces a different JP-system that serves as a generalization of the Sokolov's form of a vectorial NLS. [ABSTRACT FROM AUTHOR]